# Tagged Questions

Nullology is the study of empty sets in a set theoretic context.

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### How is it possible for a singleton to exist if ∅ is a subset of every set?

The question arises from the following statements: * " $\varnothing$ is a subset of every set. This fact (that $\varnothing \subseteq A$ for any A) is "vacuously true" (...) " (Enderton - Elements ...
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### Empty intersection and empty union

If $A_\alpha$ are subsets of a set $S$ then $\bigcup_{\alpha \in I}A_\alpha$ = "all $x \in S$ so that $x$ is in at least one $A_\alpha$" $\bigcap_{\alpha \in I} A_\alpha$ = "all $x \in S$ so that ...
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### Is $\{\}\;, \{\{\}\}\;\;,\{\{\{\}\}\}$ is an empty set or not?

Is $\{\}, \, \{\{\}\},\, \{\{\{\}\}\}$ is an empty set or not? My opinion: the first is the empty set because it contains no elements, while the second and third are not an empty set because it ...
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### Is the empty set a power set?

I have some home work with problems such as... Determine whether each of these sets is the power set of a set: {∅, {a}} ∅ So yes 1 is the power set of {a}. But what about 2? Since power sets ...
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### Powerset of set with empty sets

I am working on a problem with the following set: $$S = \{\varnothing,\{\varnothing\}\}$$ My solution was: $P(S) = \{\varnothing, \{\varnothing\},\{ \varnothing , \{ \varnothing \}\}\}$, but the ...
Currently taking a logic class and trying to understand this. You have two set $A$ and $B$. Both sets are empty sets. Is set $A$ a subset of the complement of set $B$? Assume the context is the ...
The empty set is a member of $P({a,b}) \times P({p,q})$. True or false? My first instinct was false, since the empty set is a member of each power set individually, but when multiplied together, you ...